This statement can be derived from first principles by noting that under certain conditions, the function Z depends only on the thermodynamic beta to the minus-one-half, and evaluating the derivative of the log of that function with respect to beta.  Debye's theory of heat capacity reduces to the Dulong-Petit law in the hig
This statement can be derived from first principles by noting that under certain conditions, the function Z depends only on the thermodynamic beta to the minus-one-half, and evaluating the derivative of the log of that function with respect to beta.  Debye's theory of heat capacity reduces to the Dulong-Petit law in the high-temperature limit as a direct consequence of this statement. The truth of this statement depends on the truth of the (*) ergodic hypothesis, and this statement is a direct consequence of the fact that in a Maxwell-Boltzmann distribution, the numerator of the exponent is the kinetic energy of the molecule. For ten points, identify this theorem which at equilibrium allocates energy equal to one-half times Boltzmann's constant times the temperature for every quadratic degree of freedom.
This statement can be derived from first principles by noting that under certain conditions, the function Z depends only on the thermodynamic beta to the minus-one-half, and evaluating the derivative of the log of that function with respect to beta.  
{equipartition} theorem